Tiling systems were introduced by D. Giammaresi and A. Restivo. They looked for a possibility to define recognizable picture languages by extending the projection of local languages in the one-dimensional case to two dimensions. 

Let $p \in \Sigma^{*, *}$ be a picture. We call $B_{2, 2}(\hat{p})$ the set of tiles of the picture p. To recall from the preliminaries, $B_{2, 2}(\hat{p})$ contains any sub-picture of the bordered picture $\hat{p}$ of size $(2, 2)$. We can now restate the definition of two-dimensional local languages from~\cite{cherubini2009picture}: 

\begin{definition}
	A picture of size $(2, 2)$ is a tile. Let $\Theta$ be a finite set of tiles over $\Sigma \cup \{\#\}$. A language $L \subseteq \Sigma^{*, *}$ is \emph{local}, if $L$ is generated by $\Theta$ as follows: \[L = \{p \in \Sigma^{*, *} \mid B_{2, 2}(\hat{p}) \subseteq \Theta\}\]
	
	We write $L = LOC(\Theta)$. 
\end{definition}

The family of local languages is the set of languages which can be characterized by a finite set of tiles. 

To underline the power of local languages we recapitulate a suitable example from~\cite{giammarresi1997twodimensional}. 

\begin{example}
\label{example:local_language}
	We have the following finite set of tiles. 
	\begin{align*} \Theta = \left\lbrace
	\begin{tabular}{c}
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	\# & \# \\
	\hline
	\# & 1 \\
	\hline
	\end{tabular},
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	1 & \# \\
	\hline
	\# & \# \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	\# & \# \\
	\hline
	0 & \# \\
	\hline
	\end{tabular},
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	\# & 0 \\
	\hline
	\# & \# \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	\# & 1 \\
	\hline
	\# & 0 \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	\# & 0 \\
	\hline
	\# & 0 \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	0 & \# \\
	\hline
	0 & \# \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	0 & \# \\
	\hline
	1 & \# \\
	\hline
	\end{tabular}\\[2ex]
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	0 & 0 \\
	\hline
	\# & \# \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	0 & 1 \\
	\hline
	\# & \# \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	\# & \# \\
	\hline
	1 & 0 \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	\# & \# \\
	\hline
	0 & 0 \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	1 & 0 \\
	\hline
	0 & 1 \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	0 & 1 \\
	\hline
	0 & 0 \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	0 & 0 \\
	\hline
	1 & 0 \\
	\hline
	\end{tabular}, 
	\begin{tabular}{|m{0.3cm}|m{0.3cm}|}
	\hline
	0 & 0 \\
	\hline
	0 & 0 \\
	\hline
	\end{tabular}
	\end{tabular}
	\right\rbrace
	\end{align*}
	
	The first four tiles are the only tiles which can be used in the four corners. Thus, top-left and bottom-right corners are always containing 1's whereas the other corners are always containing 0's. The following eight tiles are for vertical and horizontal borders. As can be seen, these tiles do not allow any other 1's except those in the appropriate corners. The last four tiles are forcing the picture to have 1's on the diagonal and 0's otherwise. 
	
	Therefore $LOC(\Theta) = \{p \in \{0, 1, \#\}^{**} \mid l_1(p) = l_2(p) \text{ and } p(i, i) = 1 \text{ and } p(i, j) = 0 \forall i, j \in \{1, \dots, l_1(p)\}, i \neq j\}$ is the language of square-shaped bordered pictures with 1's on the diagonal and the rest are 0's. 
\end{example}

As this example underlines, the generative power of local languages is restricted, for example it is impossible to generate square-shaped pictures with a one-letter alphabet. 

Due to the fact that LOC is the natural extension of string local languages~\cite{cherubini2009picture}, we can extend the definition of one-dimensional regular languages to the two-dimensional case:

\begin{definition}
	The quadruple $T = (\Sigma, \Gamma, \Theta, \pi)$ is called \emph{tiling system} (TS), where 
	
	\begin{compactitem}
		\item $\Sigma$ and $\Gamma$ are two finite alphabets, 
		\item $\pi: \Gamma \rightarrow \Sigma$ is a mapping and
		\item $\Theta$ is a finite set of tiles over $\Gamma \cup \{\#\}$
	\end{compactitem}
	
	The language recognized by $T$ is $L(T) = \pi(LOC(\Theta))$
\end{definition}

The family of languages generated by tiling systems is called $\familyOf{TS}$. This definition solves the restriction of shaping pictures with one letter. 

\begin{example}
	Let $T = (\Sigma, \Gamma, \Theta, \pi)$ be a tiling system with $\Sigma = \{a\}$, $\Gamma = \{0, 1\}$, $\Theta$ from Example~\ref{example:local_language} and $\pi(0) = \pi(1) = a$. 
	
	It is obvious that this tiling system is recognizing the language $L(T) = \{p \in \{a\}^{*, *} \mid l_1(p) = l_2(p)\}$, containing square pictures of a's. 
\end{example}

Before we consider tiling systems as a candidate for regular two-dimensional languages we will modify the size of the tiles. 